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{PSTYLE "Heading 2" -1 318 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 255 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 17 "C\341lculo I-MAT1610" }}{PARA 18 "" 0 "" {TEXT -1 13 "Laboratorio \+ 2" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 5 "" 0 "" {TEXT 295 7 "Martes " }{TEXT 307 10 "6 de abril" } {TEXT 308 9 " de 2010 " }{TEXT 303 94 " (aniversario de la publicaci \363n de \"El Principito\". Su lanzamiento fu\351 invisible a los oj os)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 298 16 "P lazo de entrega" }{TEXT 299 40 " : Martes 20 de abril a las 13:00 hora s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 302 30 "Pr \363ximo Laboratorio se sube el" }{TEXT -1 5 " : " }{TEXT 304 37 "Ma rtes 20 de abril a las 13:30 horas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 318 "" 0 "" {TEXT 499 10 "Importante" }{TEXT 500 79 " : La pr \363xima semana NO hay laboratorios, para preparar mejor la I1 (ahora \+ s\355!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 311 "" 0 "" {TEXT -1 0 "" }{TEXT 290 23 "NOMBRE ALUMNO DIGI TADOR" }{TEXT 296 1 ":" }}{PARA 316 "" 0 "" {TEXT -1 0 "" }{TEXT 291 0 "" }{TEXT 292 27 "N\372mero de alumno digitador:" }}{PARA 0 "" 0 "" {TEXT 293 19 "Secci\363n de C\341tedra:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 19 "NOM BRE OTRO ALUMNO:" }}{PARA 317 "" 0 "" {TEXT -1 17 "N\372mero de alumno :" }}{PARA 317 "" 0 "" {TEXT -1 19 "Secci\363n de C\341tedra:" }} {PARA 315 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 1 "(" } {TEXT 300 61 "El alumno digitador se alterna de laboratorio en laborat orio)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {TEXT -1 0 "" }}{PARA 311 "" 0 "" {TEXT 297 16 "NOMBRE DEL GRUPO" }{TEXT -1 1 ": " }}{PARA 0 "" 0 "" {TEXT 294 33 "Secci\363n de Laboratorio del grupo: " }}{PARA 255 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Objetivos" }}{PARA 0 "" 0 "" {TEXT -1 33 "Estudiar sucesiones usando Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Instrucciones" }}{PARA 15 "" 0 "" {TEXT -1 119 "Entreg ue el laboratorio sin output alguno y sin gr\341ficos. Para ello elja \+ la opci\363n Edit->Remove Output->From Worksheet." }}{PARA 15 "" 0 "" {TEXT -1 345 "Para corregir su laboratorio el ayudante ejecutar\341 pr imero el laboratorio con Edit->Execute->Worksheet. El ayudante entre gar\341 el laboratorio corregido sin output, pero con los comentarios \+ correspondientes. Los comandos deben estar en secuencia l\363gica de \+ modo de que, al ejecutarlos en orden, los c\341lculos sean correctos. \+ Para simplificar use " }{TEXT 309 7 "restart" }{TEXT -1 86 " en cada p roblema y cargue nuevamente los paquetes que necesita (with(plots), et c...) " }}{PARA 15 "" 0 "" {TEXT -1 63 "Todos los ejercicios a ser rea lizados por usted est\341n en color " }{TEXT 311 4 "blue" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 62 "La parte verbal de sus repuestas \+ debe ser entregada en color " }{TEXT 310 9 "dark red." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Glosario de Comandos Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 336 23 "Asignaci\363n a \+ variables:" }{TEXT -1 1 " " }{TEXT 0 3 ":= " }{TEXT -1 2 ", " }{TEXT 0 8 "unassign" }{TEXT -1 3 " , " }{TEXT 0 7 "restore" }{TEXT -1 2 ", \+ " }{TEXT 0 6 "assume" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 0 22 "var := comando maple; " }}{PARA 0 "" 0 "" {TEXT -1 81 "S e evalua el comando maple y la expresi\363n que resulta es asignada a la variable " }{TEXT 0 3 "var" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 12 "assume(x>a);" }{TEXT -1 1 " " }{TEXT 0 19 "assume(n,integer); " }{TEXT -1 1 " " }{TEXT 0 14 "assume(z,real)" }} {PARA 0 "" 0 "" {TEXT -1 55 "Son ejemplos de assume. Para mayor inform aci\363n ejecute " }{TEXT 0 8 "?assume;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 17 "unassign('var'); " }}{PARA 0 "" 0 "" {TEXT -1 52 "\"borra\" todo lo que se conoce acerca de la variable " } {TEXT 0 3 "var" }{TEXT -1 56 ", incluyendo las restricciones impuestas con el comando " }{TEXT 337 6 "assume" }{TEXT -1 107 ". De este modo, ella puede ser usada sin restricciones m\341s adelante. Las comillas \+ ' ' son fundamentales. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 0 8 "restore;" }}{PARA 0 "" 0 "" {TEXT -1 125 "reiniciali za todas las variables que se hayan ocupado \"borrando\" todo lo que s e conoce sobre ellas. Es equivalente a aplicar " }{TEXT 0 15 "unassign ('var')" }{TEXT -1 17 " a cada variable " }{TEXT 309 3 "var" }{TEXT -1 21 " que se haya ocupado." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "?as sume" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT 312 25 "Aproximaciones decimales:" } {TEXT -1 1 " " }{TEXT 0 5 "evalf" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 11 "Digits:= N;" }}{PARA 0 "" 0 "" {TEXT -1 90 "Define que de \+ ahora en adelante se ocupa aritm\351tica decimal con N d\355gitos sign ificativos. " }{TEXT 313 32 "Al iniciar Maple Digits vale 10" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 13 "evalf( exp r) " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 120 "Eval\372a la expr esi\363n en artim\351tica decimal con N d\355gitos sgniticativos, dond e N es el valor que tiene la variable Digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 14 "evalf(expr,N) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Evalua la expresi\363n " }{TEXT 0 4 "expr" }{TEXT -1 52 " en aritm\351tica decimal con N d\355gitos sign ificativos." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 " " {TEXT 314 41 "Gr\341fico de funciones y curvas en el plano" }{TEXT -1 2 ", " }{TEXT 0 4 "plot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 21 "plot( f(x), x=a..b); " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 94 "Grafica a y=f(x) en el intervalo [a,b]. El rango y escala del eje Y se ajusta autom\341ticamente" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 43 "plot ( f(x) , x=a..b, scal ing=constrained);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Idem \+ que el anterior pero las escalas en los ejes X a Y est\341n en relaci \363n 1:1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 30 "plot ( f(x) , x=a..b, y=c..d);" }}{PARA 0 "" 0 "" {TEXT -1 96 " Mu estra la porci\363n del gr\341fico de y=f(x) que yace en el rect\341 ngulo a <= x <= b, c<= y <= d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 41 "plot( f(x), x=a..b, view=[x1..x2,y1..y2]) " }}{PARA 0 "" 0 "" {TEXT -1 121 "Muestra la porci\363n del gr\341fic o de y=f(x) para a <= x <= b pero enmarcado en una ventana con x1 < x < x2, y1 < y < y2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 44 "plot ( f(x) , x=a..b, y=c..d, discont=true);" }}{PARA 0 "" 0 "" {TEXT -1 91 " Idem que el anterior, pero se usa cuando f es \+ discont\355nua (y f no toma valores complejos)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 55 "plot( f(x), x=a..b, y=c..d, discont=true, color=COLOR);" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 123 "Idem que el anterior donde y= f(x) se grafica con color COLOR, donde COLOR puede ser red, blue, gree n, cyan, magenta, etc..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 0 78 "plot( [f(x),g(x), h(x)] , x=a..b, y= c..d,discont=true, color=[red,blue,cyan]);" }}{PARA 0 "" 0 "" {TEXT -1 124 "Idem que en el anterior pero se grafican las tres funciones f, g,h al mismo tiempo en colores red,blue, cyan respectivamente." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 42 "plot( [x (t), y(t) , t=a..b], color=red ); " }}{PARA 0 "" 0 "" {TEXT -1 79 "gra fica los puntos (x(t),y(t)) en el plano XY cuando t var\355a desde a hasta b." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 63 "plot( [x(t), y(t) , t=a..b], color=red, scaling=constrained) ) ;" }}{PARA 0 "" 0 "" {TEXT -1 149 "Idem que el anterior, pero las esca las en los ejes X,Y est\341n en relaci\363n 1:1. (equivalente a presio nar el bot\363n 1:1 en la barra de men\372 del gr\341fico)." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 316 20 "M\341s sobre gr\341ficos: " }{TEXT 315 29 "with(plots), animate, display" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 0 12 "with(plots) " }{TEXT -1 97 "Activa al paquete plots donde \+ se encuentran las rutinas, display, implicitplot, textplot, etc...." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 55 "animate( \+ \{ f(x), g(x,a)\}, x=x1..x2, a=a1..a2, nframes);" }{TEXT -1 163 " Cr ea una animaci\363n donde en cada cuadro se grafica a las funciones f( x), g(x,a) en el intervalo [x1,x2], con a variando desde a1 hasta a2 \+ con un incremento de " }{XPPEDIT 18 0 "(a2-a1)/(nframes-1);" "6#*&,&%# a2G\"\"\"%#a1G!\"\"F&,&%(nframesGF&F&F(F(" }{TEXT -1 16 ". En total h ay " }{TEXT 319 7 "nframes" }{TEXT -1 10 " cuadros. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 26 "p1:= plot( f(x), x=a..b) : " }{TEXT -1 23 " Guarda en la variable " }{TEXT 0 2 "p1" }{TEXT -1 72 " el gr\341fico de y=f(x), el cual puede ser desplegado mediante el comando " }{TEXT 317 13 "display(p1). " }{TEXT -1 87 "Note que la \+ \372nica manera en que el gr\341fico puede ser desplegado es mediante \+ el comando " }{TEXT 318 9 "display. " }{TEXT -1 39 "Es conveniente ter minar el comando con " }{TEXT 0 1 ":" }{TEXT -1 11 " en vez de " } {TEXT 0 1 ";" }{TEXT -1 30 " para evitar ouput indeseable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 19 "display(p1,p2,p3); " }{TEXT -1 49 " Grafica los gr\341ficos guardados en las variables \+ " }{TEXT 320 9 "p1,p2,p3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 0 74 "display( sucesi\363n o lista de de variables con gr \341ficos, insequence=true); " }{TEXT -1 72 "Muestra cuadro a cuadro l os gr\341ficos en la lista creando una animaci\363n. " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 " " 0 "" {TEXT 322 68 "Estructuras de datos, (expresi\363n) sucesi\363n, lista, conjunto, tabla: " }{TEXT 321 53 "[ s], \{s\}, nops(s) , op(s) , seq(s) , s[n] , map(f,s)" }}{PARA 0 "" 0 "" {TEXT -1 153 "Para agru par varios datos en una misma variable, se puede usar una de las sigu ientes estructuras de datos: (expresi\363n) sucesi\363n, lista, conjun to, tabla." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 22 "exp1, exp2, ... , expn" }{TEXT -1 12 " Es la " }{TEXT 334 8 " sucesi\363n" }{TEXT -1 229 " formada por las expresiones exp1, exp2, \+ etc. En general, expresiones separadas por comas definen a un sucesi \363n. Las expresiones pueden ser de distintos tipos. Sucesiones respe tan el orden de sus elementos y aceptan repeticiones" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 4 "[s] " }{TEXT -1 8 " Es l a " }{TEXT 335 5 "lista" }{TEXT -1 42 " formada por los elementos de l a sucesi\363n " }{TEXT 323 1 "s" }{TEXT -1 17 ". En general, si " } {TEXT 324 2 "s " }{TEXT -1 25 "es una sucesi\363n entonces " }{TEXT 325 3 "[s]" }{TEXT -1 79 " es una lista. Listas respetan el orden de s us elementos y aceptan repeticiones" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 0 4 "\{s\} " }{TEXT -1 56 "Es el conjunto forma do por los elementos de la sucesi\363n " }{TEXT 326 1 "s" }{TEXT -1 17 ". En general, si " }{TEXT 327 2 "s " }{TEXT -1 26 "es una sucesi \363n. entonces " }{TEXT 328 3 "\{s\}" }{TEXT -1 153 " es un conjunto. En maple un conjunto emula al concepto matem\341tico de conjunto: ele mentos repetidos se eliminan y el orden de los elementos no importa. . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 15 "T[algo ]:=valor;" }{TEXT -1 146 " Si T no es una lista , crea la tabla T con una entrada. Con asignaciones adicionales del tipo T[indice]=valor se agregan elementos a la tabla T." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 5 "s[k] " }{TEXT -1 43 "el elemento k-\351simo de la lista o conjunto " }{TEXT 331 2 "s," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 7 "T[algo]" }{TEXT -1 59 " Si T es una tabla, valor asociado a \"algo\" en la tabla." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 8 "nops(s) " }{TEXT -1 46 "e l n\372mero de elementos de la lista o conjunto " }{TEXT 329 1 "s" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 6 "op(s) " }{TEXT -1 55 " la sucesi\363n con los elementos \+ del conjunto o lista " }{TEXT 330 31 "s (le quita los par\351ntesis a \+ s)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 21 "seq( a(n), n=n1..n2 )" }{TEXT -1 52 " la sucesi\363n a(n1), a(n1+1), a(n1 +2), .... a(n2). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 28 "[ seq ( (a(n), n=n1..n2 ) ] " }{TEXT -1 54 "la lista con l os elementos a(n1), a(n1+1), ... , a(n2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 28 "\{ seq ( (a(n), n=n1..n2 ) \} " } {TEXT -1 93 " el conjunto con los elementos a(n1), a(n1+1), ... , a(n2 ). Elementos repetidos se eliminan. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 9 "map(f,s) " }{TEXT -1 58 "la lista o conju nto que se obtiene de aplicar una funci\363n " }{TEXT 332 2 "f " } {TEXT -1 39 "a cada elemento de la lista o conjunto " }{TEXT 333 2 "s \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Ejempl os:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sucesion:=1,4,2,10,9,1,-4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lista:= [sucesion];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "conjunto:= \{sucesion\};" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "op(conjunto); op(lista);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "nops(lista) , nops(conjunto);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "f:= x-> x^2; map(f,lista);map(f,conjunto);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "seq( k^2,k=1..5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "datos:= [ [1,2], [3,4], [5,6 ]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "datos[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "datos[2][1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Una tabla consiste en un a asociaci\363n entre valores de un \355ndice y expresiones. Las tabla s se definen en forma din\341mica. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "T[1]:= 3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "T[4]:= 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "T[cabeza_de_p escado]:= cola_de_pescado;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "op(T);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 4 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 339 12 "Los comandos" }{TEXT -1 1 " " }{TEXT 0 12 "fsolve,solve" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 0 28 "fsolve( f(x)=g(x), x=a..b); " }}{PARA 0 "" 0 "" {TEXT -1 46 "Calcula por metodos num\351ricos UNA soluci\363n de " } {TEXT 340 9 "f(x)=g(x)" }{TEXT -1 58 " en el intervalo [a,b], CON LA E XCEPCI\323N del caso en que " }{TEXT 341 10 "f(x), g(x)" }{TEXT -1 34 " sean polinomios, donde encuentra " }{TEXT 342 24 "TODAS las raic es reales" }{TEXT -1 22 " en el intervalo [a,b]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 23 "fsolve( f(x)=g(x), x=a)" } {TEXT -1 47 " busca por m\351todos num\351ricos UNA solucion de " } {XPPEDIT 18 0 "f(x)=g(x)" "6#/-%\"fG6#%\"xG-%\"gG6#F'" }{TEXT -1 27 " \+ comenzando la b\372squeda en " }{TEXT 343 3 "x=a" }{TEXT -1 55 ". No n ecesariamente encuentra la soluci\363n m\341s cercana a" }{TEXT 344 4 " x=a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 20 "solve( f(x)=g(x), x)" }{TEXT -1 6 ". Si " }{TEXT 345 5 "f, g " }{TEXT -1 248 "son polinomios, encuentra todas las raices, e n otros casos intenta encontrar tantas soluciones como pueda mediante \+ reducciones algebraicas. Si no encuentra soluciones ya sea por que no \+ hay o por que no pudo hallarlas, retorna NULL, es decir \"nada\"." }} {PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Ej emplo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eq := x^4-5*x^2+6*x=2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(eq,x);" }}} {PARA 0 "" 0 "" {TEXT -1 62 "N\363tese que el 1 aparece dos veces: est o se debe a que es una " }{TEXT 346 10 "ra\355z doble" }{TEXT -1 73 " de la ecuaci\363n: esto es, que el polonomio en cuesti\363n es divisi ble por " }{XPPEDIT 18 0 "(x-1)^2" "6#*$,&%\"xG\"\"\"F&!\"\"\"\"#" }} {PARA 0 "" 0 "" {TEXT -1 44 "Podemos formar una lista con las solucion es:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sols := [solve(eq,x)] ;" }}}{PARA 0 "" 0 "" {TEXT -1 73 "Y ahora referirnos a una cualquiera de ellas por su posici\363n en la lista:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sols[3];" }}}{PARA 0 "" 0 "" {TEXT -1 44 "Nos entrega \+ el tercer elemento de la lista. " }}{PARA 0 "" 0 "" {TEXT -1 54 "Podem os tambi\351n verificar que la soluci\363n es correcta:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(x= sols[3],eq);" }}}{PARA 0 "" 0 "" {TEXT -1 52 "No parece tan evidente q ue ambos lados sean iguales:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(subs(x=sols[3],eq)) ;" }}}{PARA 0 "" 0 "" {TEXT -1 14 " Y s\355 lo eran!" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 352 21 "Algebra: los com andos" }{TEXT -1 1 " " }{TEXT 0 7 "expand," }{TEXT -1 1 " " }{TEXT 0 51 "combine, simplify, factor, normal, coeff, quo, rem." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 19 "expand( expresion);" }}{PARA 0 "" 0 "" {TEXT -1 139 "Expande la expresi\363n desarrollando \+ potencias de binomios, distribuyendo productos con sumas, o expandiend o f\363rmulas trigonom\351tricas, etc.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 20 "combine( expresion);" }}{PARA 0 "" 0 "" {TEXT -1 64 "aplicado a ciertas expresiones realiza la operaci \363n inversa de " }{TEXT 351 6 "expand" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 20 "simplify(expresion); " }}{PARA 0 "" 0 "" {TEXT -1 294 "aplica las reglas de simplificaci \363n que cumplen las funciones t\355picas trigonom\351tricas, expone nciaci\363n, logaritmos, polinomios, etc.. La noci\363n que Maple tien e de una expresi\363n simplificada es con toda probabilidad diferente \+ de la que Ud. tiene, la que es a su vez diferente de la de su vecino. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 18 "factor (polinomio);" }}{PARA 0 "" 0 "" {TEXT -1 74 "factoriza el polinomio en factores con coeficientes enteros , racionales. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 23 "factor(polinomio,real);" } }{PARA 0 "" 0 "" {TEXT -1 96 "factoriza el polinomio en factores con c oeficeintes reales usando aritm\351tica de punto flotante. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 26 "factor(polinomio ,complex);" }}{PARA 0 "" 0 "" {TEXT -1 90 "factoriza el polinomio en f actores lineales complejos usando aritm\351tica de punto flotante." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 18 "normal(e xpresion);" }}{PARA 0 "" 0 "" {TEXT -1 92 "obtiene una expresi\363n en la forma denominador/numerador, con t\351rminos comunes simplificados ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 23 "coeff ( expresion ,x^k);" }}{PARA 0 "" 0 "" {TEXT -1 97 "obtiene el coeficie nte de x^k en la expresion, siempre que \351sta sea una suma de ponten cias de x. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 32 "q:= quo(a,b,x); r:= rem(a,b,x);" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Si " }{TEXT 0 3 "a,b " }{TEXT -1 17 " son polinomios, " }{TEXT 0 1 "q" }{TEXT -1 20 " es el cuociente de " }{TEXT 0 1 "a" }{TEXT -1 14 " dividido por " }{TEXT 309 1 "b" }{TEXT -1 3 " y " }{TEXT 0 1 "r" }{TEXT -1 16 " es el resto \+ de " }{TEXT 0 1 "a" }{TEXT -1 14 " dividido por " }{TEXT 309 1 "b" } {TEXT -1 37 ". El resto y el couciente satisfacen " }{TEXT 0 8 "a= b*q +r" }{TEXT -1 28 ", donde grado(r) < grado (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Problemas" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " } {TEXT 432 10 "Problema 1" }{TEXT -1 44 " (Sucesiones y sus l\355mites. Versi\363n gr\341fica)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 "" {TEXT -1 127 "El t\351rmino n-\351simo de una sucesi\363n puede definirse p or una regla o funci\363n que asigna al n\372mero n el valor a(n) , po r ejemplo," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:= n -> (n^2 + 1)/n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "b:= n-> (3+n)/n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c:= n-> (1.01)^(n^2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Hasta aqu\355 tenemos simplemente fu nciones, y Maple no sabe que " }{TEXT 357 2 " n" }{TEXT -1 53 " repre senta un entero positivo. Mediante el comando " }{TEXT 353 6 "seq( )" }}{PARA 0 "" 0 "" {TEXT -1 152 "podemos transformar sus valores en una secuencia (sucesi\363n de puntos), en el sentido Maple. As\355 podemo s tambi\351n impirmirla (\277qu\351 sucede si intenta usar " }{TEXT 358 4 "plot" }{TEXT -1 51 " para graficar alguna de las sucesiones ant eriores?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq( a(n),n=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq( b(n),n=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq( c(n),n=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Para visu alizar los valores de una sucesi\363n se pueden graficar los puntos ( " }{XPPEDIT 18 0 "n,a[n]" "6$%\"nG&%\"aG6#F#" }{TEXT -1 38 ") en el pl ano para ciertos valores de " }{TEXT 354 1 "n" }{TEXT -1 65 ". Para el lo se forma una lista con los pares ordenados [n,a(n)] ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "pares _ordenados:= [seq([n,b(n)],n=1..10)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Cuando se grafican pares \+ de puntos se elige la opci\363n " }{TEXT 355 1 " " }{TEXT 400 11 "sty le=point" }{TEXT -1 61 " para que los puntos no sean conectados por una poligonal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "p1:=plot(lista_de_pares,style=point,symbol=cir cle):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 " ahora graficamos" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "display(p1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 356 43 "\241Cuidado con la interpretacion del g r\341fico!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "En el ejemplo anterior pareciera que la sucesi\363n se aproxim a a cero, sin embargo " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 " " 0 "" {XPPEDIT 18 0 "b[n] = 3/n+1;" "6#/&%\"bG6#%\"nG,&*&\"\"$\"\"\"F '!\"\"F+F+F+" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "evidentemente " }} {PARA 310 "" 0 "" {XPPEDIT 18 0 "1 < b[n];" "6#2\"\"\"&%\"bG6#%\"nG" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Podemos evitar el error anterior usando la opci\363n " }{TEXT 401 19 "scaling=constrained" }{TEXT -1 48 " , claro que a costa de la \+ est\351tica del gr\341fico." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "display(p1,scaling=constrain ed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Por \372ltimo menciona mos que Maple puede calcular l\355mites de ciertas expresiones media nte el comando " }{TEXT 434 5 "limit" }{TEXT -1 38 " (busque ayuda s obre dicho comando) :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(b(n),n=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 347 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "1i)" }}{PARA 0 "" 0 "" {TEXT 350 23 "Considere la sucesi\363n " }{XPPEDIT 402 0 "a[n] = 2^n/(3^(n+1)); " "6#/&%\"aG6#%\"nG*&)\"\"#F'\"\"\")\"\"$,&F'F+F+F+!\"\"" }{TEXT 360 108 " . Liste los primeros 10 elementos de dicha sucesi\363n. \277Qu \351 observa? Repita con los siguientes 10 elementos." }}{PARA 0 "" 0 "" {TEXT 359 17 "Ahora liste los " }{XPPEDIT 403 0 "a[n];" "6#&%\"aG6 #%\"nG" }{TEXT 361 9 " desde " }{XPPEDIT 404 0 "n = 100;" "6#/%\"nG \"$+\"" }{TEXT 362 8 " hasta " }{XPPEDIT 405 0 "n = 110;" "6#/%\"nG\" $5\"" }{TEXT 363 76 ". Grafique cada una de las tres listas generadas \+ (tres gr\341ficos diferentes)." }}{PARA 0 "" 0 "" {TEXT 406 0 "" }} {PARA 0 "" 0 "" {TEXT 407 67 "A partir de tales observaciones conjetur e el l\355mite de la sucesi\363n." }}{PARA 0 "" 0 "" {TEXT 410 0 "" }} {PARA 0 "" 0 "" {TEXT 348 9 "Respuesta" }{TEXT 349 1 ":" }}{PARA 0 "" 0 "" {TEXT 411 0 "" }}{PARA 0 "" 0 "" {TEXT 412 0 "" }}}{PARA 0 "" 0 " " {TEXT 413 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 414 4 "1ii)" }}{PARA 0 "" 0 "" {TEXT 415 0 "" }}{PARA 0 "" 0 "" {TEXT 416 63 "Repita los pe dido en la pregunta anterior para la sucesi\363n " }{XPPEDIT 417 0 "a[n] = sqrt(n^2+n)/(2*n+1);" "6#/&%\"aG6#%\"nG*&-%%sqrtG6#,&*$F'\" \"#\"\"\"F'F/F/,&*&F.F/F'F/F/F/F/!\"\"" }}{PARA 0 "" 0 "" {TEXT 418 0 "" }}{PARA 0 "" 0 "" {TEXT 408 9 "Respuesta" }{TEXT 409 1 ":" }}{PARA 0 "" 0 "" {TEXT 419 0 "" }}{PARA 0 "" 0 "" {TEXT 420 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 421 0 "" }{TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "1iii)" }}{PARA 0 "" 0 "" {TEXT 424 41 "Repita otra vez, ahora para la sucesi\363n " }{TEXT 426 1 " " }{XPPEDIT 427 0 "a [n] = sqrt((n+1)*(n+2))-n;" "6#/&%\"aG6#%\"nG,&-%%sqrtG6#*&,&F'\"\"\"F .F.F.,&F'F.\"\"#F.F.F.F'!\"\"" }}{PARA 0 "" 0 "" {TEXT 425 0 "" }} {PARA 0 "" 0 "" {TEXT 422 9 "Respuesta" }{TEXT 423 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "1iv)" }}{PARA 0 "" 0 "" {TEXT 430 33 "Y una vez m\341s \+ para la sucesi\363n " }{XPPEDIT 431 0 "a[n]=(2/3)^n/(1-n^(1/n))" "6# /&%\"aG6#%\"nG*&)*&\"\"#\"\"\"\"\"$!\"\"F'F,,&F,F,)F'*&F,F,F'F.F.F." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 428 9 "Respuest a" }{TEXT 429 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 433 10 "Problema 2" }{TEXT -1 35 " (Algebra d e l\355mites de sucesiones)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 437 172 "Calcule los 4 l\355mites de la Pregunta 1 \+ \"a mano\", usando las t\351cnicas de \341lgebra de l\355mites aprendi das en clase. Finalmente, confirme todos los l\355mites usando el coma ndo " }{TEXT 536 5 "limit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 435 9 "Respuesta" }{TEXT 436 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 438 10 "Problema 3" }{TEXT -1 39 " (L\355mite s de sucesiones por definici\363n)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 " " {TEXT -1 76 "Lo visto en los ejercicios anteriores indica que, a me dida que n crece ( " }{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$% )operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 8 ") los " } {XPPEDIT 18 0 "a[n] " "6#&%\"aG6#%\"nG" }{TEXT -1 60 " se parecen m \341s y m\341s a L, en el sentido que la desigualdad " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "abs(a[n]-L) < epsilon;" "6#2-%$absG6#,&&%\"aG6#%\"nG \"\"\"%\"LG!\"\"%(epsilonG" }{TEXT -1 30 " puede cumplirse para todo " }{XPPEDIT 18 0 "epsilon " "6#%(epsilonG" }{TEXT -1 31 " a contar \+ de alg\372n valor de " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 25 " ( el que depender\341 del " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" } {TEXT -1 14 " escogido). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Si, dada una sucesi\363n " }{XPPEDIT 18 0 "a[n ]" "6#&%\"aG6#%\"nG" }{TEXT -1 22 " , existe un n\372mero " } {XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 28 " tal que la desigualdad \+ " }{XPPEDIT 18 0 "abs(a[n]-L) < epsilon" "6#2-%$absG6#,&&%\"aG6#%\"nG \"\"\"%\"LG!\"\"%(epsilonG" }{TEXT -1 31 " se cumple a contar de alg \372n " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 24 " , cualquiera sea \+ el " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 25 " escogi do, decimos que " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 8 " es el \+ " }{TEXT 439 6 "l\355mite" }{TEXT -1 17 " de la sucesi\363n " } {XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 17 " y escribimos \+ " }{XPPEDIT 18 0 "L=limit(a[n],n=infinity)" "6#/%\"LG-%&limitG6$&%\"aG 6#%\"nG/F+%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 " NOTA: El " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 50 " a contar del cual se verifica la desigualdad " }{XPPEDIT 18 0 "abs(a[n]-L) < ep silon" "6#2-%$absG6#,&&%\"aG6#%\"nG\"\"\"%\"LG!\"\"%(epsilonG" }{TEXT -1 31 " depender\341, normalmente del " }{XPPEDIT 18 0 "epsilon" "6 #%(epsilonG" }{TEXT -1 10 " usado. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Si mi ramos la desigualdad " }{XPPEDIT 18 0 "abs(a[n]-L) < epsilon" "6#2-% $absG6#,&&%\"aG6#%\"nG\"\"\"%\"LG!\"\"%(epsilonG" }{TEXT -1 26 " como una inecuaci\363n en " }{TEXT 443 1 "n" }{TEXT -1 82 " , necesitamos que su soluci\363n ( o parte de ella) sea un intervalo de la forma ] " }{TEXT 442 1 "a" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "infinity" "6#%)in finityG" }{TEXT -1 39 " [ , lo cual implicar\341 que para todo " } {TEXT 440 1 "n" }{TEXT -1 21 " mayor o igual que " }{TEXT 441 1 "a" }{TEXT -1 28 " se cumplir\341 con lo pedido." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Por ejemplo, para verifi car que " }{XPPEDIT 18 0 "limit((2*n+5)/(3*n+2),n=infinity)=2/3" "6# /-%&limitG6$*&,&*&\"\"#\"\"\"%\"nGF+F+\"\"&F+F+,&*&\"\"$F+F,F+F+F*F+! \"\"/F,%)infinityG*&F*F+F0F1" }{TEXT -1 39 " necesitamos resolver \+ la inecuaci\363n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "abs((2*n+5)/(3*n+2)-2/3) " 0 "" {MPLTEXT 1 0 20 "restart :with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=n->(2*n +5)/(3*n+2); epsilon:=0.07;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(abs(f(n)-2/3) " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Maple no sabe que " }{TEXT 445 1 "n" }{TEXT -1 166 " de nota un n\372mero natural y resuelve la inecuaci\363n como si se trat ase de un n\372mero real. De all\355 el segundo intervalo de la soluci \363n. Si lo ignoramos tenemos que " }{XPPEDIT 18 0 "abs(f(n)-2/3)=17" "6#1\"#<%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "Ello significa que para tales " }{TEXT 446 1 "n" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "2/3-epsilon " 0 "" {MPLTEXT 1 0 73 "franja:=plot(\{2/3-epsilon,2/3+epsi lon\},x=10..30,linestyle=2,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "puntos:=plot([[n,f(n)]$n=10..30],style=point,symbol=c ircle,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "displa y(franja,puntos,view=[9..31,0..1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "y observamos que a partir de n=17 los puntos quedan de ntro de la franja." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Para establecer con certeza que " }{XPPEDIT 18 0 "limit ((2*n+5)/(3*n+2),n=infinity)=2/3" "6#/-%&limitG6$*&,&*&\"\"#\"\"\"%\"n GF+F+\"\"&F+F+,&*&\"\"$F+F,F+F+F*F+!\"\"/F,%)infinityG*&F*F+F0F1" } {TEXT -1 35 " debemos poder hacer lo anterior " }{TEXT 447 12 " para todo " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT 448 10 " posi tivo" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "unassign('epsilon');" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 98 "# con ello epsilon deja de tener el valor a signado previamente y vuelve a ser una variable libre." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(abs(f(n)-2/3) " 0 "" {MPLTEXT 1 0 77 "# Maple no sabe que epsilo n representa un n\372mero positivo. Se lo indicamos:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume(epsilon>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(abs(f(n)-2/3) " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Y vemos que si " }{XPPEDIT 18 0 "n[0]" "6#&%\"nG6#\"\"!" }{TEXT -1 33 " es el menor entero mayo r que " }{XPPEDIT 18 0 "11/(9*epsilon)-6/9" "6#,&*&\"#6\"\"\"*&\"\"*F &%(epsilonGF&!\"\"F&*&\"\"'F&F(F*F*" }{TEXT -1 23 " entonces, para to do " }{XPPEDIT 18 0 "n>=n[0] " "6#1&%\"nG6#\"\"!F%" }{TEXT -1 15 " se satisface " }{XPPEDIT 18 0 "abs(f(n)-2/3)<2/3" "6#2-%$absG6#,&-%\"fG 6#%\"nG\"\"\"*&\"\"#F,\"\"$!\"\"F0*&F.F,F/F0" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Veamos el valor de " }{XPPEDIT 18 0 "11/(9*epsilon)-6/9" "6#,&*&\"#6\"\"\"*& \"\"*F&%(epsilonGF&!\"\"F&*&\"\"'F&F(F*F*" }{TEXT -1 32 " para distin tos valores de " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 4 " : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "x0:=11/(9*epsilon)-6/9 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(epsilon=0.07,x0);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "#el mismo que hab\355amos hallado antes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(epsilon=0.0002,x0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(epsilon=0.000003,x0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "As\355, " }{XPPEDIT 18 0 "abs(f(n)-2/3)" "6#-%$absG6#,&-%\"fG6#%\"nG\"\"\"*&\"\"#F+\"\"$!\"\"F/ " }{TEXT -1 37 " < 0.0002 para todo n>=6111 y " }{XPPEDIT 18 0 "abs(f(n))" "6#-%$absG6#-%\"fG6#%\"nG" }{TEXT -1 35 " < 0.000003 pa ra todo n>=407407" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "El hecho que podamos hallar un " }{XPPEDIT 18 0 "n[0]" "6#&%\"nG6#\"\"!" }{TEXT -1 24 " apropiado para cada " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 43 " positivo establece que \+ el l\355mite es 2/3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "3i)" }}{PARA 0 "" 0 "" {TEXT 451 35 "Resuelva \" a m ano \" la inecuaci\363n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {TEXT 453 2 " " }{XPPEDIT 477 0 "abs((2*n+5)/(3*n+2)-2/3)=n[0] " "6#1&%\"nG6#\"\"!F% " }{TEXT 480 13 " se verifique" }}{PARA 0 "" 0 "" {TEXT 481 0 "" }} {PARA 310 "" 0 "" {XPPEDIT 493 0 "abs(b[n]-L)<1/100" "6#2-%$absG6#,&&% \"bG6#%\"nG\"\"\"%\"LG!\"\"*&F,F,\"$+\"F." }}{PARA 0 "" 0 "" {TEXT 482 0 "" }}{PARA 0 "" 0 "" {TEXT 483 70 "En seguida grafique la franja horizontal delimitada por las rectas " }}{PARA 0 "" 0 "" {XPPEDIT 552 0 "y=L-1/100" "6#/%\"yG,&%\"LG\"\"\"*&F'F'\"$+\"!\"\"F*" }}{PARA 0 "" 0 "" {TEXT 484 7 " e " }}{PARA 0 "" 0 "" {XPPEDIT 553 0 "y=L+ 1/100" "6#/%\"yG,&%\"LG\"\"\"*&F'F'\"$+\"!\"\"F'" }}{PARA 0 "" 0 "" {TEXT 485 100 " junto a los puntos de la sucesi\363n en alg\372n int ervalo apropiado y verifique visualmente que los " }{XPPEDIT 494 0 " b[n] " "6#&%\"bG6#%\"nG" }{TEXT 487 33 " caen dentro de la franja par a " }}{PARA 0 "" 0 "" {XPPEDIT 495 0 "n>=n[0] " "6#1&%\"nG6#\"\"!F%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 478 27 "Luego \+ confirme su respuesta" }}{PARA 0 "" 0 "" {TEXT 464 20 "usando el coma ndo " }{MPLTEXT 1 0 5 "limit" }{TEXT 465 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 455 9 "Respuesta" }{TEXT 456 1 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "3i ii)" }}{PARA 0 "" 0 "" {TEXT 467 6 "Sean " }{XPPEDIT 496 0 "b[n]" "6# &%\"bG6#%\"nG" }{TEXT 468 5 " y " }{TEXT 466 1 "L" }{TEXT 469 37 " \+ los de la parte anterior y sea " }{XPPEDIT 18 0 "epsilon" "6#%(eps ilonG" }{TEXT 554 13 " un n\372mero " }{TEXT 555 19 "positivo arbitr ario" }{TEXT 556 33 ". Encuentre un n\372mero natural " }{XPPEDIT 497 0 "n[0] " "6#&%\"nG6#\"\"!" }{TEXT 470 23 " tal que para todo \+ " }{XPPEDIT 303 0 "n>=n[0]" "6#1&%\"nG6#\"\"!F%" }{TEXT 471 14 " se v erifique" }}{PARA 0 "" 0 "" {TEXT 472 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 305 0 "abs(b[n]-L) " 0 "" {MPLTEXT 1 0 18 "phi:= x-> x/2+1/x;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "y " }{XPPEDIT 18 0 "a[n+1] = phi( a[n] )" "6#/&%\"aG6#,&%\"nG\"\"\"F)F)-%$phiG6#&F%6#F( " }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a[1]:=2; for i from 1 to 10 do a[i+1]:=eval f(phi(a[i])); od; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "4i)" }}{PARA 0 "" 0 "" {TEXT 371 73 "De la observaci\363n de los elementos vistos arriba \277cu\341l pareciera \+ ser el " }{TEXT 372 6 "l\355mite" }{TEXT 373 41 " de la sucesi\363n a _n (el n\372mero al cu\341l " }{TEXT 374 7 "tienden" }{TEXT 375 18 " \+ dichos valores)? " }}{PARA 0 "" 0 "" {TEXT 376 8 "Cambie " }{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT 377 23 " por cualquier n\372mer o " }{TEXT 378 8 "positivo" }{TEXT 379 155 " de su elecci\363n y estud ie los primeros t\351rminos de la sucesi\363n resultante. \277Tienden \351stos al mismo l\355mite que antes? Repita, escogiendo ahora dos v alores " }{TEXT 389 9 "negativos" }{TEXT 390 6 " para " }{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT 391 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 10 "Respuesta:" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "4ii)" }}{PARA 0 "" 0 "" {TEXT 385 36 "Observe que si, efectivamente, los " }{XPPEDIT 18 0 "a_n" "6#%$a_ nG" }{TEXT 382 29 " tienden a un valor l\355mite, " }{XPPEDIT 18 0 "L " "6#%\"LG" }{TEXT 383 21 ", entonces tambi\351n " }{XPPEDIT 18 0 "l imit(a[n+1],n=infinity)= L" "6#/-%&limitG6$&%\"aG6#,&%\"nG\"\"\"F,F,/F +%)infinityG%\"LG" }{TEXT 399 130 " (pues se trata de la misma sucesi \363n, \"desplazada\" en una unidad). Usando esta idea y, encontrando \+ las ra\355ces de la ecuaci\363n " }{XPPEDIT 18 0 "phi(x)=x" "6#/-% $phiG6#%\"xGF'" }{TEXT 384 28 " , elabore un argumento que " }{TEXT 380 10 "justifique" }{TEXT 381 37 " su respuesta a la pregunta anterio r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 387 4 "Not a" }{TEXT -1 12 ": Un punto " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 17 " que satisface " }{XPPEDIT 18 0 "phi(x) = x" "6#/-%$phiG6#%\"xGF '" }{TEXT -1 15 " se llama un " }{TEXT 388 10 "punto fijo" }{TEXT -1 15 " de la funci\363n " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 386 10 " Respuesta:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 338 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "4iii) (T e\363rico)" }}{PARA 0 "" 0 "" {TEXT 394 93 "La idea anterior, de usar los puntos fijos de la funci\363n iteradora para hallar el valor de \+ " }{XPPEDIT 558 0 "L=limit(a[n],n=infinity)" "6#/%\"LG-%&limitG6$&%\"a G6#%\"nG/F+%)infinityG" }{TEXT 396 31 ", se basa en tener previamente " }{TEXT 397 11 " la certeza" }{TEXT 398 74 " de que \351ste existe. P ara asegurar que ello ocurre basta que la sucesi\363n " }{XPPEDIT 559 0 "a[n] " "6#&%\"aG6#%\"nG" }{TEXT 501 82 " se creciente y acotada superiormente, o bien decreciente y acotada inferiormente." }}{PARA 0 "" 0 "" {TEXT 395 57 "Demuestre que la sucesi\363n de las partes ant eriores (con " }{XPPEDIT 560 0 "a[1]=1" "6#/&%\"aG6#\"\"\"F'" }{TEXT 557 179 ") satisface una de esas parejas de condiciones (\277cu\341l \+ de ellas?), con lo cual queda garantizado que el l\355mite existe, y \+ \351ste debe, entonces, ser el valor hallado en la parte i)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 393 10 "Respuest a:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 548 10 "Problema 5" }{TEXT -1 37 " (Definici \363n de limites de funciones)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 " " {TEXT -1 33 "El comportamiento de la funci\363n " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " , cuando " }{TEXT 517 1 "x" } {TEXT -1 14 " se ecerca a " }{TEXT 518 1 "a" }{TEXT -1 16 " se donot a por " }{XPPEDIT 18 0 " limit(f(x),x=a)" "6#-%&limitG6$-%\"fG6#%\"xG/ F)%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Decimos que " }{XPPEDIT 18 0 "L=limit(f(x),x=a)" "6 #/%\"LG-%&limitG6$-%\"fG6#%\"xG/F+%\"aG" }{TEXT -1 45 " si, dado cua lquier intervalo alrededor de " }{TEXT 519 1 "L" }{TEXT -1 12 ", diga mos " }{XPPEDIT 18 0 "I[epsilon]" "6#&%\"IG6#%(epsilonG" }{TEXT -1 3 "=] " }{XPPEDIT 18 0 "L-epsilon,L+epsilon" "6$,&%\"LG\"\"\"%(epsilonG! \"\",&F$F%F&F%" }{TEXT -1 44 " [ , puede hallarse intervalo alrededor de " }{TEXT 520 1 "a" }{TEXT -1 15 ": " }{XPPEDIT 18 0 " I[delta]" "6#&%\"IG6#%&deltaG" }{TEXT -1 3 "=] " }{XPPEDIT 18 0 "a-del ta,a+delta" "6$,&%\"aG\"\"\"%&deltaG!\"\",&F$F%F&F%" }{TEXT -1 30 " [, talq que todos los puntos " }{XPPEDIT 18 0 " x<>a" "6#0%\"xG%\"aG" } {TEXT -1 5 " en " }{XPPEDIT 18 0 " I[delta] " "6#&%\"IG6#%&deltaG" } {TEXT -1 4 " ( " }{TEXT 521 4 "si " }{XPPEDIT 256 0 "0 \+ " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a:=1; epsilon:=0.05;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "Observamos el gr\341fico de f(x) cerc a de x=a:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(f(x),x=a-0.5. .a+0.5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " creemos que " }{XPPEDIT 18 0 "limit(f(x),x = 1) = 1;" "6#/-%&lim itG6$-%\"fG6#%\"xG/F*\"\"\"F," }{XPPEDIT 18 0 "`` = L;" "6#/%!G%\"LG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Graficamente esta es la situaci\363n:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "curva:=plot(f(x),x=a-0.5..a+0.5,thickness=2):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 128 "franja1:=plot(L+epsilon,x=a-0.5..a+0.5,thickn ess=2,color=green):franja2:=plot(L-epsilon,x=a-0.5..a+0.5,thickness=2, color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "display(cu rva,franja1,franja2,view=[a-0.5..a+0.5,L-2*epsilon..L+2*epsilon]);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "La franj a horizontal representa el intervalo " }{XPPEDIT 18 0 "I[epsilon]" " 6#&%\"IG6#%(epsilonG" }{TEXT -1 16 ". Para hallar " }{XPPEDIT 18 0 " delta" "6#%&deltaG" }{TEXT -1 62 " geomet\351tricamente, buscamos las intersecciones de la curva " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG 6#%\"xG" }{TEXT -1 29 " con las l\355neas horizontales " }{XPPEDIT 18 0 "y = 1+epsilon;" "6#/%\"yG,&\"\"\"F&%(epsilonGF&" }{TEXT -1 5 " , \+ " }{XPPEDIT 18 0 "y = 1-epsilon;" "6#/%\"yG,&\"\"\"F&%(epsilonG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Simplemente haciendo click en los puntos en que la curva (en " }{TEXT 533 4 "rojo" }{TEXT -1 32 ") entra y sale de la franja ( en " }{TEXT 534 5 "verde" }{TEXT -1 181 ") y observando las coordenad as del dichos puntos (en la esquina superior izquierda) vemos que tale s puntos son (aprox.) (0.98, 0.95) y (1.02, 1.05), lo cual indica r\355a que si " }{XPPEDIT 18 0 "abs(x-a)<``" "6#2-%$absG6#,&%\"xG\"\" \"%\"aG!\"\"%!G" }{TEXT -1 8 "0.02 = " }{XPPEDIT 18 0 "delta" "6#%&de ltaG" }{TEXT -1 18 ", se cumple que " }{XPPEDIT 18 0 "abs(f(x)-L) " 0 "" {MPLTEXT 1 0 78 "x_izq:=fsolve(f(x)=L-epsilon,x=0.5..1);x_der:=fsolve(f(x)=L+epsi lon,x=1..1.5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Luego, si elegimos delta = min\{ x_der -a , a - x_izq), se verificar\341 lo \+ deseado:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "delta:=min( x_der-a,a -x_izq);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "v1:=plot([a-del ta,y,y=L-2*epsilon..L+2*epsilon],color=black):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "v2:=plot([a+ delta,y,y=L-2*epsilon..L+2*epsilon],color=black):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 84 "display(curva,franja1,franja2,v1,v2,view=[a- 0.15..a+0.15,L-2*epsilon..L+2*epsilon]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Obviamente cuaquier delta menor \+ que el escogido tambi\351n serv\355a." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "Tambi\351n podemos analizar el proble ma as\355:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Si, para cierto delta elegimos " }{XPPEDIT 18 0 "x" "6#%\"xG " }{TEXT -1 11 " tal que " }{XPPEDIT 18 0 "abs(x-1) " 0 "" {MPLTEXT 1 0 22 "solve(x^2+2*x<0.05,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 29 "Y notamos que se cumple con " } {XPPEDIT 18 0 "abs(f(x)-L)